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This paper considers the design of online power allocation strategies for slotted energy harvesting point-to-point communications systems. The objective is to minimize the competitive rate gap that is defined as the maximum gap between the optimal rates that can be achieved by the offline and online policies over all possible energy arrival profiles. A lower-bound on the competitive rate gap is derived and a new online policy is presented. The new online policy is shown to perform very close to the competitive rate gap lower-bound for any number of slots and to outperform previously proposed online policies, such us the myopic policy.

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